This paper is organized as follows. In Section 2 we give necessary preliminaries, including the mathematical description of one-way LOCC in terms of operator relations. Section 3 includes a brief introduction to the relevant operator structures in our analysis, and then we present some of our main results and applications from [1]}, [2]}, [3]}. We finish in Section 4 by first giving a brief introduction to our necessary notions from graph theory, then we present one of our main results from [4]} and some examples, and we derive a new graph-theoretic description for the case of a single qubit sender.
